Module 5 Lecture - Non-parametric Comparisons for More than Two Groups

Analysis of Variance

Quinton Quagliano, M.S., C.S.P

Department of Educational Psychology

1 Overview and Introduction

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

1.1 Learning Objectives

2 Continuing Mean Comparisons

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

2.1 A Priori / Planned / Orthogonal Contrasts

  • A priori / planned / orthogonal comparisons are used to test specific contrast hypotheses established prior to collecting the data.

  • Contrasts are orthogonal if the tests are independent of one another: that is if the outcome of one contrast cannot predict the outcome of the other contrast. Two contrasts are orthogonal if they account for non-overlapping between group variance.

  • Two contrasts are orthogonal if the sum of the cross-product of the corresponding coefficients of these contrasts equals zero:

    • \(\sum{(C_{j1})(C_{j2})} = 0\) where:
      • \(C_{j1}\) weight given to mean \(j\) in 1st contrast
      • \(C_{j2}\) weight given to mean \(j\) in 2nd contrast
  • Important: In k groups, there are only k-1 contrasts orthogonal to each other
Contrast G1 G2 G3 G4
A -1 -1 1 1
B -3 1 1 1
C -1 1 -1 1
  • Examples of checking cross-sums:
    • Are contrast tests A and B orthogonal?:
      • \([(-1)(-3)] + [(-1)(1)] + [(1)(1)] + [(1)(1)] = 4 \rightarrow\) non-orthogonal
    • Are contrast tests A and C orthogonal?:
      • \([(-1)(-1)] + [(-1)(1)] + [(1)(-1)] + [(1)(1)] = 0 \rightarrow\) orthogonal
  • Discuss: Are contrasts B and C orthogonal? Work through the math to show this

Scheffe A Priori

The Bonferroni Inequality

Dunn-Sidak

2.2 A Posteriori / Post-hoc

  • For the following post-hoc tests, we do not need to assess whether they are orthogonal or not

Scheffe Post Hoc

Tukey HSD

Dunnett’s Test for Treatment/Control Comparisons

3 Conclusion

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

3.1 Recap

3.2 Lecture Check-in

  • Make sure to complete any lecture check-in tasks associated with this lecture!

Module 5 Lecture - Non-parametric Comparisons for More than Two Groups || Analysis of Variance